Lloyd can paint a room in 15 hours. Pablo can paint the same room in 9 hours. How long does it take for both Lloyd and Pablo to paint the room if they are working together?

Write an equation using the variable  to represent the time it takes to paint the room together.

Equation:    

How long does it take?     hours

Let's solve the problem step by step.

### Step 1: Understand the Rates of Work

- **Lloyd's rate** of painting is \( \frac{1}{15} \) of the room per hour.
- **Pablo's rate** of painting is \( \frac{1}{9} \) of the room per hour.

### Step 2: Combine Their Rates

If they work together, their combined rate of painting the room is the sum of their individual rates:

\[
\text{Combined rate} = \frac{1}{15} + \frac{1}{9}
\]

Let \( t \) represent the time (in hours) it takes for both of them to paint the room together. The equation that represents this situation is:

\[
\frac{1}{15}t + \frac{1}{9}t = 1
\]

### Step 3: Solve the Equation

First, find a common denominator for the fractions \( \frac{1}{15} \) and \( \frac{1}{9} \). The least common denominator (LCD) of 15 and 9 is 45.

Convert the fractions:

\[
\frac{1}{15} = \frac{3}{45}, \quad \frac{1}{9} = \frac{5}{45}
\]

Now, substitute these into the equation:

\[
\frac{3}{45}t + \frac{5}{45}t = 1
\]

Combine the fractions:

\[
\frac{8}{45}t = 1
\]

Now, solve for \( t \) by multiplying both sides by the reciprocal of \( \frac{8}{45} \):

\[
t = \frac{45}{8} = 5.625 \text{ hours}
\]

### Final Answer

It takes **5.625 hours** (or 5 hours and 37.5 minutes) for Lloyd and Pablo to paint the room together.

Would you like more details or have any questions about this solution?

### Follow-Up Questions:
1. How much of the room would Lloyd paint alone in 5 hours?
2. If Pablo took a 2-hour break, how long would it take them to paint the room together?
3. How long would it take if a third person could paint the room in 6 hours and joined them?
4. If they painted two rooms together, how long would it take?
5. What if Lloyd worked twice as fast—how would that change the time to paint the room together?

### Tip:
When combining work rates, add the fractions that represent each worker's rate and then solve for the total time.

Discover how long it takes for Lloyd and Pablo to paint a room together. Solve the problem step-by-step with detailed explanations of rates of work and equations with fractions. Suitable for Grades 7-9.

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